## Linear Operators: General theory |

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Page 28

1 : D → X be a generalized sequence of elements in a metric space X. We call f a

generalized Cauchy sequence in X , if , for each x > 0 , there

such that elf ( p ) , f ( q ) ) < € if p 2 do , q 2 do . 5 LEMMA . If f is a generalized ...

1 : D → X be a generalized sequence of elements in a metric space X. We call f a

generalized Cauchy sequence in X , if , for each x > 0 , there

**exists**a do € D ,such that elf ( p ) , f ( q ) ) < € if p 2 do , q 2 do . 5 LEMMA . If f is a generalized ...

Page 292

Then lim un ( E )

for which lim Mm ( EF )

in E , for which EF € E , for each F € E2 . It is clear that if F , and F , are elements ...

Then lim un ( E )

**exists**for EE . n - 00 PROOF . Let E , be the family of all sets Einfor which lim Mm ( EF )

**exists**for each F € Ej , and let Ez be the family of all sets Ein E , for which EF € E , for each F € E2 . It is clear that if F , and F , are elements ...

Page 362

Under the hypotheses of Exercise 37 , show that there

n ( x ) dx if and only if the functions commenco mnanon ( ) , m 2 1 , are uniformly

bounded and equicontinuous . 89 Let { an } , -00 < n < too , be a bounded ...

Under the hypotheses of Exercise 37 , show that there

**exists**f in C with S * 4.x ) *n ( x ) dx if and only if the functions commenco mnanon ( ) , m 2 1 , are uniformly

bounded and equicontinuous . 89 Let { an } , -00 < n < too , be a bounded ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero